Cup product in bounded cohomology of negatively curved manifolds

TL;DR

This paper investigates the structure of bounded cohomology in negatively curved manifolds, showing that classes from exact forms lie in the radical of the cup product, revealing new algebraic properties.

Contribution

It demonstrates that classes from exact differential 2-forms in bounded cohomology are in the radical of the cup product, extending previous results from hyperbolic surfaces to higher dimensions.

Findings

Classes from exact forms are in the radical of the cup product.

Extension of Barge and Ghys' results to negatively curved manifolds.

Provides new insights into the algebraic structure of bounded cohomology.

Abstract

Let $M$ be a negatively curved compact Riemannian manifold with (possibly empty) convex boundary. Every closed differential $2$-form $\xi\in\Omega^2(M)$ defines a bounded cocycle $c_\xi\in C_b^2(M)$ by integrating $\xi$ over straightened $2$-simplices. In particular Barge and Ghys proved that, when $M$ is a closed hyperbolic surface, $\Omega^2(M)$ injects this way in $H_b^2(M)$ as an infinite dimensional subspace. We show that any class of the form $[c_\xi]$, where $\xi$ is an exact differential 2-form, belongs to the radical of the cup product on the graded algebra $H_b^\bullet(M)$.